A.1 Distillation Process

A. Design Example

Reference:[SP05] S. Skogestad and I. Postlethwaite,

Multivariable Feedback Control; Analysis and Design,Second Edition, Wiley, 2005.

[SP05, Sec. 13.4]

2

Models: Understand the ProcessSTEP 1. Real Physical System

STEP 2. Ideal Physical Model

STEP 3. Ideal Mathematical Model

STEP 4. Reduced Mathematical Model

Conceptual/Schematic model（図式化・概念化）

Idealization（理想化）

Linearization（線形化）

Product（本物）

Gas

Liquid

BubbleTray

Stage

3

STEP 2. Ideal Physical Model

Reflux flow rate, kmol/min.Boilup from reboiler, kmol/min.

Bottom product rate, kmol/min.Distillate (top product) rate, kmol/min.

Inputs

Outputs

Number of stages: 40

4

STEP 3. Ideal Mathematical Model

Tray ’s composition dynamics can be formulated as follows:

where

Liquid holdup on theoretical tray , kmol.

Liquid mole fraction of light component on stage .

Vapor mole fraction of light component on stage .

Feed

Overheadvapor Reflux

Boilup

Bottom Flow

5

STEP 4. Reduced Mathematical Model

Then, the whole system can be viewed as a first-order model.

Each tray has its own physical model.

Feed

Overheadvapor Reflux

Boilup

Bottom Flow

Assumptions

• The flow dynamics are immediate.• All trays have the same dynamic responses.

Distillation Process: Problem Statement

6

Real Physical System Ideal Physical Model[SP05, pp. 100, 509-514]

: top composition: bottom composition

: reflux: boilup

Controlled Variables

Manipulated Inputs: distillate

: bottom flow: overhead vapor

Assumption • The composition dynamics are usually much slower than the flow dynamics

the simplifying assumption of perfect control of hold up and instantaneous flow responses in the column

Flow RelationshipsTop/Bottom CompositionInputs Outputs

2-Input 2-Output System

7

Distillation Process: Plant Model [SP05, pp. 100, 509-514]

influences

Nominal Model

MATLAB CommandN = {87.8, -86.4; 108.2 -109.6};D = [75 1];Pnom = tf(N, D);

Gain Margin :Delay Margin :

Multiplicative (Output) Uncertainty

Distillation Process

Nominal Model

Uncertain Plant Model

[min]

9

Rise time 30 min

Uncertainty Weight

[rad/min]

0.035Gain Crossover

Frequency

1 min

1 rad/min

Performance Weight

Distillation Process: Performance Specifications

rad/min

rad/min

Steady state error < 0.01

rad/min

Delay Margin:

1.0

Gain Margin: 20%, 2dB

10

Multivariable Feedback Control

-

-

Multi-Input Multi-Output(MIMO) System

How to design multivariable feedback controllers systematically?

Non-interactionSingle-Input Single-Output(SISO) System -

-

11

Distillation Process: SISO Plant Model

Plant

Pole:

Zero: none

Stable System

Minimum Phase System

Re

Im

Frequency Response (Bode Plot) Step Response

0

0.1

0

Frequency [rad/min]

12

Distillation Process: SISO Plant Model

Plant:Re

Im

Frequency Response (Bode Plot) Step Response

0

0.1

0

Pole:

time scaling rad/min rad/s

Frequency [rad/min]

13

Auto PID tuning algorithm: pidtuneDistillation Process: SISO Controller Design

K = pidtune( Pnom, ‘pidf‘ ) ;L = series( K, P ) ;T = feedback( L, 1 ) ;figure; bode( L ) ;figure; step( T ) ;

PID:

Bode diagram

Step response

MATLAB Command

14

-controller

Bode diagram Step response

Perturbed Plant ModelNominal Plant Model

(See 6th lecture)Auto tuning algorithm:

Distillation Process: SISO Plant Model

Frequency [rad/min]

-controller (update)

Bode diagram Step response

(See 6th lecture)Auto tuning algorithm:

Distillation Process: SISO Plant Model

Frequency [rad/min]

16

-controller

Bode diagram Step response

Perturbed Plant ModelNominal Plant Model

(Order 11 → 4)(See 6th lecture)

Auto tuning algorithm:Distillation Process: SISO Plant Model

Frequency [rad/min]

-controller (update)

Bode diagram Step response

(Order 11 → 4)(See 6th lecture)

Auto tuning algorithm:Distillation Process: SISO Plant Model

Frequency [rad/min]

Frequency [rad/min]Frequency [rad/min]18

Distillation Process: Evaluate SISO Controller Design*

PID

Complementary SensitivitySensitivity

0.02241 rad/min0.0227 rad/min

-controller

1.01

Mag

nitu

de [d

B]

Mag

nitu

de [d

B]

19

Distillation Process: Evaluate SISO Controller Design*Complementary SensitivitySensitivity

-controller

Frequency [rad/min] Frequency [rad/min]

Mag

nitu

de [d

B]

Mag

nitu

de [d

B]

20

Control of Multivariable Plants1. Diagonal Controller (decentralized control)

[SP05, pp. 91-93]

-

-

Controller

0

0.1

0

00

Nominal Plant ModelTime delay

0 1.0

NSNP

RPRS

○

×

NSNP

RPRS

○

×

×

×

○ ○

21

(Input Uncertainty)

2. Two-step compensator design: dynamic decoupling

-

-

Controller

0

0.1

0

00

Inverse-based controller (decoupling control)

Nominal Plant ModelTime delay

0 1.0

NSNP

RPRS

○

×

×

○

NSNP

RPRS

○

×

×

○

22

Inverse-based controller (decoupling control)2. Two-step compensator design: dynamic decoupling

(proper) (proper)

NSNP

RPRS

×

×

×

×

NSNP

RPRS

×

×

×

×

0

0.1

0

00

-

-

23

-

Multivariable Feedback Control

(Input Uncertainty)

controller

NSNP

RPRS

○

○

×

○

24

Control of Multivariable Plants1. Diagonal Controller (decentralized control)

[SP05, pp. 91-93]

-

-

Controller

0

0.1

0

00

Nominal Plant ModelTime delay

0 1.0

NSNP

RPRS

○

×

NSNP

RPRS

○

×

○

○

○

○

Performance Weight

25

-

Multivariable Feedback Control

(Input Uncertainty)

controller

NSNP

RPRS

○

○

○

○

NSNP

RPRS

○

○

○

○

26

Another Example [2]

[2] R.K. Wood and M.W. Berry, “Terminal composition control of a binary distillation column,”Chemical Engineering Science, Vol. 28, No. 9, pp. 1707-1717, 1973.

Column’s diameter: 9inThe number of tray: 8The space of each tray: 12in4 bubble caps are arranged in a square patternand each size is in

(Above distillation column is interfaced with an IBM 1800)A total condenser and basket type reboiler is equipped.

Then, the above distillation process’s model was determined as followsfrom its step response.

where

Distillation of methanol